This thesis is a study of the integration of proximity functions over certain
compact groups. Mean values are found of the ultrametric valuation of certain
rational functions associated with a divisor on an abelian variety, and it is shown
how these may be expressed in terms of an integral, thus finding the analogue, for
an abelian variety, of Mahler's definition of the measure of a polynomial. These
integrals are shown to arise in a manner which mimics classical Riemann sums, and
their relation with the global canonical height is investigated. It is shown that the
measure is a rational multiple of log p. Similar results are given for elliptic curves,
taking the divisor to be the identity of the group law, and somewhat stronger
mean value theorems proven in this more specific case by working directly with
local canonical heights rather than approaching them through related functions.
Effective asymptotic formulae for the local height are derived, first for the kernel of
reduction of a curve and then, via a detailed analysis of the local reduction of the
curve, for the group of rational points. The theory of uniform distribution is used to
show that the mean value also takes an integral form in the case of an archimedean
valuations, and recent inequalities for elliptic forms in logarithms are used to give
error terms for the convergence towards the measure. This is undertaken first for the
local height on an elliptic curve, and then, in terms of general theta-functions, on an
abelian variety. We then seek to exploit these generalisations of the Mahler measure
to yield an alternative method to that of Silverman and Tate for the determining of
the global height. The integration over a cyclic group of the laws satisfied locally by
the height allows us to reformulate our theorems in a manner conducive to practical
application. It is demonstrated how our asymptotic formulae may be used together
with an appropriate computer software package, PARI in our case, to calculate the
mean value of heights, and, more generally, of rational functions, on an elliptic curve
and on abehan varieties of higher genus. Some such calculations are displayed, with
comments on their efficacy and their possible future development.